Lawson’s three criteria
These conditions are very severe…
J. D. Lawson, 1955
In 1955 a young engineer working on nuclear fusion decided to work out exactly how enormous the task of achieving fusion is. Although his colleagues were optimistic about their prospects, he wanted to prove it to himself. His name was John Lawson, and his findings – that the conditions for fusion power relied on three vital quantities – became the landmark Lawson Criteria.
The genesis of Lawson’s Criteria is simple enough – he calculated the requirements for more energy to be created than is put in, and came up with a dependence on three quantities: temperature (T), density (n) and confinement time (τ). With only small evolution thanks to some subtle changes of definition, this is basically the same figure of merit used by today’s fusion scientists – the triple product, nτT.
The amount of energy created relies on particles colliding and fusing – the number of collisions is related to the number of particles in a certain region – thus n, the number density (not mass density) is Lawson’s first criterion. This would seem encouraging for the prospective experiment, as creating high pressure is relatively easy. However there is a catch. At higher densities a process known as bremsstrahlung rears its ugly head, in which collisions between nuclei and electrons generate radiation. Bremsstrahlung can become so dominant that all the power in the plasma is radiated away – the optimum density conditions are surprisingly low, around a million times less dense than air.
Nonetheless the fusion collisions – between the nuclei - have to be at high speed. This allows the nuclei to overcome their electrostatic repulsion, and get close enough for the strong force – that governs fusion – to take over and stick the particles together. The speed of a gas or plasma particle is equivalent to its temperature: the second of Lawson’s criteria. Again there is a limit – if the two particles are moving really fast then the time they are in close enough proximity for fusion to occur decreases. The bremsstrahlung also increases at higher temperatures, due to faster moving electrons. The Goldilocks temperature turns out to be in the vicinity of 100 – 200 million degrees, a seemingly huge task in the fifties that has become a standard condition today.
With the first two criteria satisfied fusion reactions can occur, but to get a substantial amount of power generated you need time to allow the reactions to happen – on this basis Lawson’s third criteria comes into play, the energy confinement time. This is the time that energy remains in the plasma before escaping, and it is here that the most remarkable gains have been made during the course of fusion research. From only microseconds in Lawson’s time, the confinement time has improved by a factor of a million to reach about one second in JET and is planned to hit around 5 seconds in ITER.
It is a neat quirk of physics that conditions for achieving ignition in a tokamak depend linearly on these three quantities – just multiply them together to get an indication of your tokamak’s performance. The magic figure for ignition is a triple product of greater than 5 x 1021 m-3s keV . In its high power experiments of 1997 JET achieved around one fifth of that value.
Thanks to the years of study of fusion, our knowledge of how to achieve these criteria is very detailed. An appropriate value for density is easily achieved and tokamak heating systems have successfully created the necessary temperatures with radio-frequency heating and neutral beam injection. The remaining piece of the puzzle is the confinement time. Great improvements have been made in the plasma profiles and magnetic control that influence the confinement, but the largest improvement can be made simply by building a bigger tokamak, thereby increasing the plasma volume relative to its surface area.
John Lawson died in 2008, and so did not live to see ITER turn on. However, with its ten-fold increase over JET’s volume, the huge fusion experiment will put the last piece into place and see Lawson’s criteria finally satisfied.